The even and odd Hilbert transformations, H+ and H−, are known to be bounded from the power-weighted space ℒμp to itself for −1 < μ 1 and 0 < μ < 2 respectively. We show that they fail to be surjective on ℒ0,p and ℒ1, p respectively, and we characterise the spaces H+(ℒ0, p), H- (ℒ1, p) and find inverses for H+, H− on them.

We treat the time-harmonic Maxwell equations in an exterior domain with prescribed boundary data [n, E] in the Sobolev space of square integrable tangential fields with square integrable surface divergence. By using boundary integral equation methods, existence and uniqueness results are established. Furthermore, we investigate the completeness of electric and magnetic dipoles distributed on an inner surface in this Sobolev space.

This paper is a continuation of [2], where we introduced the notion of global k-spreads on manifolds. Here we show that the space of all k-spreads on a manifold has the structure of an affine space, modelled on the vector space of sections of a certain vector bundle. We give some sufficient conditions for a manifold admitting an integrable k-spread to be a space of constant curvature and answer one of the questions raised in [2].

The paper deals with smooth nonlinear ODE systems in ℝn, ẋ = f(x), such that the derivative f′(x) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f′(x(t))y. Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝn is then explicitly constructed.

We present a new result on the existence of periodic solutions for the equation:

for all positive parameters λ sufficiently large. Our fundamental assumption is the following monotonicity property: if ø ≧ ψ (ø and ψ are data) then x(ø) ≧ x(ψ). The proof consists in applying the global Hopf bifurcation theorem. The main steps are: (i) a classical estimation of the periods; (ii) an a priori estimate for the solutions along a connected branch; (iii) a transformation acting on periodic solutions.

In this paper we consider the global existence (in time) of the Cauchy problem of the semilinear wave equation utt – Δu = F(u, Du), x ∊ Rn, t > 0. When the smooth function F(u, Du) = O((|u| + |Du|)k+1) in a small neighbourhood of the origin and the space dimension n > ½ + 2/k + (1 + (4/k)2)½/2, a unique global solution is obtained under suitable assumptions on initial data. The method used here is associated with the Lorentz invariance of the wave equation and an improved Lp–Lq decay estimate for solutions of the homogeneous wave equation. Similar results can be extended to the case of “fully nonlinear wave equations”.

We study a boundary integral equation method for transmission problems for strongly elliptic differential operators, which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with Galerkin's procedure. The method is shown to work for the vector Helmholtz equation in ℝ3 with electromagnetic transmission conditions. We propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over H1. We analyse the boundary integral equations using the calculus of pseudodifferential operators. Here the concept of the principal symbol is used to derive existence and regularity results for the solution.

A general decomposition theorem that allows one to express uniquely arbitrary differential polynomials in one independent and one dependent variable as a combination of conservative, dissipative and higher order dissipative pieces is proved. The decomposition generalises the Rayleigh dissipation law for linear equations.

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set xα−1 either contains x or contains an element of (im α)′.

Write rank α for |im α| and fix α for |{x ∊ N: xa = x}|. Define (x, xα, xα2) to be an admissible α-triple if x ∊ (im α)′, xα3 ≠ xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if

A subset Y of a free completely regular semigroup FCRx freely generates a free completely regular subsemigroup if and only if (i) each -class of FCRx contains at most one element of Y, (ii) {Dy;y ∊ Y} freely generates a free subsemilattice of the free semilattice FCRx/), and (iii) Y consists of non-idempotents. A similar description applies in free objects of some subvarieties of the variety of all completely regular semigroups.

A 3-dimensional autonomous ordinary differential equation is studied which models certain cellular biochemical reactions. Extended Poincaré-Bendixson theory is used to obtain algebraic conditions on the parameters which are sufficient for the existence of at least one stable closed trajectory. Similar conditions are also obtained for the absence of chaos and for the global convergence of solutions to a critical point.